Title |
Numerical Algebraic Geometry and Algebraic Numerical Analysis |

Organizers |
Michele Bolognesi et Daniele A. Di Pietro |

Goals |
The goal of this seminar is to explore recent developments at the intersection of algebraic geometry and numerical analysis |

Format |
Two-three sessions per years with internationally recognized speakers |

Acknowledgements |
ANR projects HHOMM, fast4hho, NEMESIS and industrial collaborations with EDF |

**Morse-theoretic signal compression and reconstruction**

In this lecture I will present work of three of my PhD students, Stefania Ebli, Celia Hacker, and Kelly Maggs, on cellular signal processing. In the usual paradigm, the signals on a simplicial or chain complex are processed using the combinatorial Laplacian and the resultant Hodge decomposition. On the other hand, discrete Morse theory has been widely used to speed up computations, by reducing the size of complexes while preserving their global topological properties.

Ebli, Hacker, and Maggs have developed an approach to signal compression and reconstruction on chain complexes that leverages the tools of algebraic discrete Morse theory,, which provides a method to reduce and reconstruct a based chain complex together with a set of signals on its cells via deformation retracts, preserving as much as possible the global topological structure of both the complex and the signals. It turns out that any deformation retract of real degreewise finite-dimensional based chain complexes is equivalent to a Morse matching. Moreover, in the case of certain interesting Morse matchings, the reconstruction error is trivial, except on one specific component of the Hodge decomposition. Finally, the authors developed and implemented an algorithm to compute Morse matchings with minimal reconstruction error, of which I will show explicit examples.

**Linear PDE with Constant Coefficients**

We discuss practical methods for computing the space of solutions to an arbitrary homogeneous linear system of partial differential equations with constant coefficients. These rest on the Fundamental Principle of Ehrenpreis--Palamodov from the 1960s. We develop this further using recent advances in computational commutative algebra.

**Compatible finite element spaces for metrics with curvature**

The operators of linear elasticity can be arranged in complexes, and a goal is to construct subcomplexes consisting of finite element spaces. Such complexes are related to de Rham complexes through a diagram chase known as the BGG construction. The construction of finite element spaces makes appear cochains with coefficients in rigid motions. A de Rham theorem on cohomology groups can be proved in the case without curvature, whereas a Bianchi identity can be proved for the curvature. This is joint work with Kaibo Hu.

**Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra**

In this work, merging ideas from compatible discretisations and polyhedral methods, we construct novel fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra. The spaces and operators that appear in these sequences are directly amenable to computer implementation. Besides proving exactness, we show that the usual three-dimensional sequence of trimmed Finite Element spaces forms, through appropriate interpolation operators, a commutative diagram with our sequence, which ensures suitable approximation properties. The discrete de Rham (DDR) sequence is then used to design a stable arbitrary-order approximation of a magnetostatics problem.

**Introduction and applications of Numerical Algebraic Geometry**

*Part I: Introduction to Numerical Algebraic Geometry.* Nonlinear polynomial equations have been solved for several millennia such as those documented on Babylonian clay tablets involving the relationship between perimeter and area of rectangles. The Abel-Ruffini theorem and the development of Galois theory two centuries ago showed that solutions to most systems of polynomial equations could not be expressed in terms of radicals necessitating development of numerical computational methods to approximate solutions. This talk will explore various numerical methods for computing and analyzing solutions to systems of polynomial equations, collectively called numerical algebraic geometry. Some recent results for polynomial systems arising in science and engineering applications along with current computational challenges associated with solving polynomial systems will be discussed.

*Part II: Applications of Numerical Algebraic Geometry.* Systems of nonlinear polynomial equations arise in a variety of fields in mathematics, science, and engineering. Many numerical techniques for solving and analyzing solution sets of polynomial equations over the complex numbers, collectively called numerical algebraic geometry, have been developed over the past several decades. However, since real solutions are the only solutions of interest in many applications, there is a current emphasis on developing new methods for computing and analyzing real solution sets. This talk will summarize some numerical real algebraic geometric approaches as well as recent successes of these methods for solving a variety of problems in science and engineering.

**High order Whitney forms on simplices**

Whitney elements on simplices are perhaps the most widely used finite elements in computational electromagnetics. They offer the simplest construction of polynomial iscrete differential forms on simplicial complexes. Their associated degrees of freedom (dofs) have a very clear physical meaning and give a recipe for discretizing physical balance laws, e.g., Maxwell's equations. As interest grew for the use of high order schemes, such as hp-finite element or spectral element methods, higher-order extensions of Whitney forms have become an important computational tool, appreciated for their better convergence and accuracy properties. However, it has remained unclear what kind of cochains such elements should be associated with: Can the corresponding dofs be assigned to precise geometrical elements of the mesh, just as, for instance, a degree of freedom for the space of Whitney 1-forms belongs to a specific edge? We address this localization issue. Why is this an issue? The existing constructions of high order extensions of Whitney elements follow the traditional FEM path of using higher and higher "moments" to define the needed dofs. As a result, such high order finite $k$-elements in $n$ dimensions include dofs associated to $q$-simplices, with $k < q \le n$ , whose physical interpretation is obscure. The present presentation offers an approach based on the so-called "small simplices", a set of subsimplices obtained by homothetic contractions of the original mesh simplices, centered at mesh nodes (or more generally, when going up in degree, at points of the principal lattice of each original simplex). Degrees of freedom of the high-order Whitney k-forms are then associated with small simplices of dimension k only. We provide an explicit basis for these elements on simplices and we justify this approachfrom a geometric point of view (in the spirit of Hassler Whitney's approach, still successful 30 years after his death).

**Spline functions for geometric modeling and numerical simulation**

In geometric modeling, a standard representation of shapes is based on parameterized surfaces or volumes, which involve piecewise polynomial functions also known as splines. We will first describe these splines, in the univariate case, how they are characterized and what are their main properties. We will present some extensions to higher dimensions and how they are used for modeling shapes, in several domains of applications. We will also illustrate several exemples of spline constructions, that have been investigated. In the last decades, a new paradigm called isogeometric analysis has emerged, which uses splines functions not only to describe the geometry but also to approximate functions on this geometry. We will briefly describe how this approach is working in numerical simulation, what are its main characteristics and we will provide some illustrations. Analyzing the spaces of spline functions associated to a given domain partition or a given abstract topology is an important problem for geometric modeling and numerical simulations. We will present some of the know results and some problems still open. We will describe algebraic-geometric techniques, which provide a better insight on these spline spaces, in particular for the analysis of their dimension and for the construction of basis. This analysis involves topological complexes and homological tools, which will be explained by examples. Spline spaces over triangular meshes or T-meshes, in dimension 2 and 3, as well as extensions to geometrically smooth spline spaces (if time permits) will be considered in details.

**From the finite element method to the finite element exterior calculus**

The finite element exterior calculus (FEEC) has been developed in the early 2000s as a mathematical framework that uses the calculus of differential forms for the formulation of the finite element method. The aim of this lecture is to revisit some of the main aspects of FEEC. We shall use as a main motivation the approximation of Laplace equation in mixed form and of the time harmonic formulation of Maxwell's equations. It is remarkable that, besides providing a more elegant and comprehensive understanding of the existing theory, the FEEC allows for the development of new finite elements and for the solution of problems that would be difficult, if not impossible, to address with more traditional tools

Updated 2/12/2022